There are no approved ways to perform hardware repairs on Jetpacks other than to send it into Verizon for servicing under your warranty.
If the warranty is expired then you can feel free to experiment however you like. I've never opened up a jetpack but can imagine if you find the screws you should be able to remove the outer shell and expose the power button. From there it would be a delicate procedure to find the appropriate tools and cleaning solutions and clean whatever goop may have accumulated in that area.
If the button is just plain broken then you will likely be out of luck. To my knowledge replacement parts for Jetpacks are not available.